Question: Given $ m \angle ABC = 8x - 64$, $ m \angle CBD = 4x - 34$, and $ m \angle ABD = 58$, find $m\angle CBD$. $B$ $A$ $D$ $C$
Explanation: From the diagram, we see that together ${\angle ABC}$ and ${\angle CBD}$ form ${\angle ABD}$ , so $ {m\angle ABC} + {m\angle CBD} = {m\angle ABD}$ Substitute in the expressions that were given for each measure: $ {8x - 64} + {4x - 34} = {58}$ Combine like terms: $ 12x - 98 = 58$ Add $98$ to both sides: $ 12x = 156$ Divide both sides by $12$ to find $x$ $ x = 13$ Substitute $13$ for $x$ in the expression that was given for $m\angle CBD$ $ m\angle CBD = 4({13}) - 34$ Simplify: $ {m\angle CBD = 52 - 34}$ So ${m\angle CBD = 18}$.